The Saudi family already claimed all their land and any natural wealth discovered in the future in their land as the property of the state, the family, or all Saudi citizens, I'd bet. Perhaps they signed some profit sharing contracts with U.S. oil companies, but I doubt they signed away the rights to all oil as discovered in the future.
As to the rights of the discoverer, I think Demidog gives us a good starting point in that discussion:
I own the mineral and water rights to my property and I have no technological ability to drill for oil. Nobody may enter my property and drill for oil simply because they own the technology.